Rocking block simulation based on numerical dissipation

RESEARCH Rocking block simulation based on numerical dissipation A. M. D’Altri .G. Vlachakis .S. de Miranda .P. B. Lourenc¸o Received: 3 January 2024 / Accepted: 4 July 2024 The Author(s) 2024 Abstract In this paper, a computational approach based on numerical dissipation is proposed to simulate rocking blocks. A rocking block is idealized as a solid body interacting with its foundation through a contactbased formulation. An implicit time integration scheme with numerical dissipation, set to optimally treat dissipation in contact problems, is employed. The numerical dissipation is ruled by the time step and the rocking dissipative phenomenon at impacts is accurately predicted without any damping model. A broad numerical campaign is conducted to define a regression law in analytic form for the setting of the time step, depending on the block size and aspect ratio, the contact stiffness, as well as the coefficient of restitution selected. The so-obtained regression law appears accurate and an a posteriori validation with cases not in the training dataset confirms the effectiveness of the approach. Finally, the comparison with available experimental tests highlights the approach efficacy for free rocking and harmonic loading cases (in a deterministic sense), and for earthquake-like loading cases (in a statistical sense). It is found that rocking blocks with sizes of interest for structural engineering (e.g., cultural heritage structures) can be simulated with time steps within 10 –3 710 –1 s, so allowing very fast computations. Keywords Rocking Dynamics Masonry Out-ofplane collapse Finite element method Cultural heritage structures 1 Introduction In the last decades, rocking structures have been intensely investigated, and various models to predict the rocking motion have been developed. On the one hand, this was motivated by the need of analyzing the dynamic response of existing and cultural heritage structures, e.g., masonry and dry-stone walls [1–5], stone monuments [6–9], as well as statues [10,11], that typically experience damage/collapse due to seismic events. On the other hand, rocking structures attracted the attention of researchers as they might be used as seismic design strategies [12], given that the uplift of a rocking block limits the design forces acting in the superstructure, as well as in the foundation. This ‘‘rocking isolation’’ strategy can be used on both buildings [13] and bridges [14]. One significant issue with the seismic response of a rocking block is the large sensitivity to its defining features, i.e. minor changes in rocking block A. M. D’Altri (&)S. de Miranda Department of Civil, Chemical, Environmental, and Materials Engineering, University of Bologna, Bologna, Italy e-mail: [email protected] A. M. D’Altri G. Vlachakis P. B. Lourenc¸o Department of Civil Engineering, ISISE, ARISE, University of Minho, Guimara ˜es, Portugal 123 Nonlinear Dyn https://doi.org/10.1007/s11071-024-09974-1(0123456789().,-volV)(0123456789().,-volV) properties may result in significantly different timehistory responses. Indeed, experiments involving dynamically excited rocking specimens are rarely replicable, and the response is typically labeled as chaotic [12] (i.e., nonreproducible and unpredictable). Accordingly, a plausible approach to proceed with model validation, instead of the classical approach of comparing deterministically the specimen and the model responses under a specific ground excitation, should be based on statistical validation (as proposed in [15]). The most established model to predict the response of a rocking block has been introduced by Housner [16]. Such well-known analytical model, even though the solution is typically obtained numerically given the event-by-event formulation, represents the socalled classical rocking theory, based on the hypotheses of (i) rigid block and rigid foundation, (ii) two potential contact points, (iii) no sliding, (iv) no bouncing, and (v) energy dissipation at impacts. Based on the classical rocking theory [16], several enhancements and extensions [17–37] have been developed to treat a wide range of rocking structures with a multitude of different boundary conditions. However, the hypothesis of no sliding (as well as no bouncing) might be too strict for many actual applications, as sliding (and bouncing) is not always prevented. For this reason, more general analytical models accounting also for sliding (as well as bouncing) have been proposed [38–48]. Nonetheless, most of these models did not find widespread actual applications given the complexities and limitations in the generalization of the problem. In this context, numerical approaches may represent an appealing choice to generalize the solution for rocking problems, as they are able to deal with complex geometries, boundary conditions, and mechanical aspects (such as sliding, bouncing, 3D effects, material nonlinearities etc.). When considering masonry and cultural heritage structures, the use of block-based numerical models [49] also allows to account for the actual masonry pattern as well as the interaction with adjacent structural elements. In this framework, the adoption of contact-based numerical approaches appears particularly appropriate to model rocking blocks. The explicit time integration scheme has been typically preferred, see e.g. applications within the so-called discrete element method (DEM) [50–56]. The main drawback of these contactbased explicit approaches consists in the definition of a suitable damping model. Indeed, the choice and the characterization of a damping model (e.g., Rayleigh damping) is challenging and non-univocal [50], as the rocking block response is nonlinear, and a representative frequency of the rocking motion cannot be defined univocally. Accordingly, the setting of the damping model is mostly conducted to fit some reference response in a phenomenological fashion, rather than having a clear physical meaning. To bridge the gap between classical rocking theory and numerical models, in terms of energy dissipation, an equivalent viscous damping model calibrated on analytical solutions has been proposed in [57]. The adoption of implicit time integration schemes to model rocking blocks has found less interest in the scientific community, as such schemes are typically characterized by numerical (i.e., algorithmic) dissipation [58,59], and so the response depends on the time step chosen [60,61]. Indeed, very small time steps should be adopted to reduce the amount of numerical dissipation that would lead to inconvenient simulations. However, by noticing that when dealing with rocking motion the setting of damping models appears as questionable as relying on numerical dissipation only, this paper investigates the possibility of utilizing an implicit time integration scheme with numerical dissipation without any damping model to simulate rocking blocks. In other words, the use of numerical dissipation to account for the rocking energy dissipation does not appear more problematic than ad hoc calibrated damping models, and it leads to superior computational efficiency. The potential benefit of this choice, beyond its simplicity, is indeed immediately clear: if only numerical dissipation is employed, rather large time steps can be used, so allowing very fast and convenient computations. In this framework, a pioneering approach was proposed in [62] where rocking blocks were modelled by means of beam elements with no-tension zerolength fiber cross-sections representing the rocking surfaces, using a corotational formulation to account for geometric nonlinearity. In particular, the adoption in [62] of the well-known implicit time integration scheme with numerical dissipation proposed in [58], also known as HHT or a-method, and quasi-rigid rocking surfaces allowed to obtain classical rocking solutions through numerical dissipation only, without having a strong dependency on the adopted time step 123 A. M. D’Altri et al. (which could vary between 0.0001 s and 0.001 s), yet without directly controlling the energy dissipation. In this paper, a rocking block is idealized as a solid body interacting with its foundation through a contactbased formulation. The HHT time integration scheme is employed with an algorithmic setting to optimally treat dissipation in contact problems [63]. The rocking dissipative phenomenon at impacts is investigated, correlating its dependency on the time step. A broad numerical campaign is conducted to define a regression law in analytic form for the setting of the time step, using as reference the classical rocking theory. Comparisons with available experimental tests are used to check the efficacy of the regression law. The paper is structured as follows. Section 2 discusses the modelling assumptions at the basis of the present computational approach. Section 3presents the strategy adopted to define a suitable setting of the time step, as well as the obtained regression law together with its post validation. Section 4shows the comparison with available experimental tests, particularly free rocking and harmonic loading cases (in a deterministic sense) from [64], and earthquake-like loading cases (in a statistical sense) from [15]. 2 Modeling of rocking blocks In this section, the classical approach to model a rigid rocking block according to [16], used as reference, is firstly briefly recalled. Then, the proposed computational approach based on numerical dissipation to model a solid deformable body interacting with its foundation though a contact-based formulation is discussed together with few details about the adopted time integration method. 2.1 Classical rocking theory According to the classical rocking theory [16], the symmetric rocking block (Fig. 1) is idealized as a rigid body, characterized by the semi-diagonal Rand slenderness a¼atan B=HðÞ, rocking on a rigid foundation. The hypotheses of no sliding and no bouncing yield. The equation of motion of the rigid block in inplane free rocking about pivot points Oand O0, and measured using the rocking angle h(Fig. 1) is: € htðÞ¼p2sin asgn htðÞðÞhtðÞ½ fg ð1Þ where p¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðmgR=I0Þ pis the frequency parameter of the rigid rocking block, with mrepresenting the mass of the block, and I0representing the rotational moment of inertia with respect to the pivot points. For rectangular cuboid blocks, I0¼4=3ðÞmR2and, hence, p¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3g=4RÞ p. Impacts between the block and the foundation occur when h¼0. At any impact, the pivot point changes and the rotation changes sign. Importantly, impacts result in instantaneous energy losses. According to [16], the reduction of energy at any impact may be described using the coefficient of restitution r, defined as the ratio of postimpact to preimpact kinetic energy. Furthermore, within the preceding assumptions, the classical rocking theory [16] provided an estimation of the coefficient of restitution by employing the conservation of angular momentum: r¼1mR2 I0 1cos2aðÞ  2 ð2Þ It should be underlined that, although various more recent formulations (see e.g. [31,57,65]) adopt as coefficient of restitution ffiffir p, i.e. the ratio of the preand post-impact angular velocities, the original definition of ratio of kinetic energies is herein considered. In free rocking motion, maximum rocking angles hn and half rocking periods Tn=2 along with the number of impacts n are defined, according to [16], as: Fig. 1 The rocking block 123 Rocking block simulation based on numerical dissipation hn¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1rn11h0 ðÞ 2 hi r; Tn=2¼2 ptanh1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rn11h0 ðÞ 2 hi rð3Þ where hn¼0¼h0is the initial rocking angle, and Tn¼0=2¼T0=2 is computed by doubling the time elapsed until the first impact. In the following, Eqs. (1) and (3) are referred to as analytical solutions. As a result, the rocking behavior is nonlinear due to (i) the change of pivot point (from Oto O0and vice versa) and (ii) the jump discontinuity of the angular velocity preand post-impact caused by the impact energy dissipation, ruled by the coefficient of restitution. Further insights on the nonlinear nature of the rocking behavior can be found in [19,66–70]. 2.2 Numerical modeling A free-standing rocking block is idealized as a solid deformable body interacting with its foundation (Fig. 2a) though a contact-based formulation, characterized by a finite contact stiffness kn. The Young’s modulus of the block is assumed so that the overall block stiffness is much higher than the contact stiffness, i.e., the block Young’s modulus becomes irrelevant to the present study. Additionally, a reasonably high value of friction coefficient prevents sliding to occur. In other words, the contact stiffness knis the only mechanical parameter which has a direct and significant effect on the rocking behavior of the block. The one-step implicit time integration method with numerical dissipation developed in [58], also called HHT method (as well as a-method), is considered. The HHT method approximates the solution of an undamped structural dynamics problem by means of the following relationships: M€ uiþ1þ1þaHHT ðÞKuiþ1aHHT Kui¼Fiþ1 uiþ1¼uiþDt_ uiþDt21=2bHHT ðÞ € uiþbHHT € uiþ1 ½ _ uiþ1¼_ uiþDt1cHHT ðÞ € uiþc€ uiþ1 ½ ð4Þ Fig. 2 Proof of concept for free rocking response. aSolid block rocking on its foundation. Comparison of free rocking response in terms of bnormalized rocking angle time history, and c normalized total energy time history (being E the total energy), for a block with 2H¼4:2m,2B¼0:6m,r¼0:94, h0=a¼0:5 (the present solution has been obtained with kn¼5e ?08 N/ m 3 ,Dt¼0:013 s) 123 A. M. D’Altri et al. For i ¼0,1;2;..., being Dtthe time step, andaHHT , bHHT , and cHHT parameters governing the numerical dissipation and the stability of the algorithm (the subscript HHT has been added to avoid any confusion with other parameters). In Eq. (4), Mis the mass matrix, Kis the stiffness matrix, Fis the external forces vector, and uis the displacement vector (superimposed dots symbolize time differentiation). The solution is initiated through initial conditions u0¼uð0Þ;_ u0¼_ uð0Þ;and € u0¼M1F0ðÞKu0 ðÞ. To optimally treat numerical dissipation in the elastodynamic contact problem, the setting of aHHT , bHHT , and cHHT is carried out according to [63]. In particular, the time integration parameters are adopted to ensure unconditional stability, second-order accuracy, momentum transfer in dynamic rigid impact problems and optimal numerical dissipation, [62] as: aHHT ¼ ffiffiffi 2 pþ1¼0:41421; bHHT ¼1=2¼0:5;cHHT ¼ffiffiffi 2 p1=2¼0:91421 ð5Þ Accordingly, once defined aHHT ,bHHT , and cHHT , numerical dissipation is fully governed by the time step Dt. The higher the Dt, the higher will be the numerical dissipation. 2.3 Proof of concept The possibility to find a certain time step Dtthat guarantees good estimates of the rocking response is shown in Fig. 2. The free rocking response of a solid block (Fig. 2a) with initial rocking angle h0=a¼0:5 obtained by the present solution is compared, in terms of normalized rocking angle (Fig. 2b) and normalized total energy (Fig. 2c), with the analytical solution given by Eq. (1) and the explicit numerical solution from [57]. It can be observed that the free rocking response is accurately reproduced by the present solution and the step-like rocking dissipative phenomenon at impacts is well predicted, even without any damping model. It should be pointed out that the solution in Fig. 2 has been obtained with a rather large time step (in this case Dt¼0:013 s), which thus allowed a very fast simulation (416 s on a commercial laptop) given the limited number of increments needed. Further features of rocking response of the present solution shown in Fig. 2can be gathered by observing the phase portraits in Fig. 3. Indeed, the phase portraits show an overall good agreement between the analytical and the present solutions (see Fig. 3, left). In particular, it appears worth to highlight the response around impacts (see, e.g., the first impact in Fig. 3, top right). On the one hand, the analytical solution shows a sharp jump of the angular velocity at impact, fully governed by the coefficient of restitution. On the other hand, the present solution shows a smoother response at impacts, given the presence of a deformable contact interface. Indeed, the contact pressure distribution (see Fig. 3, bottom right) at initial condition (A) starts to sensibly change at the instant (B), i.e., the instant in which analytical and numerical solutions tend to drift apart. The first impact happens between (C) and (D), where in both cases the contact pressure is rather distributed on a considerable portion of the contact interface, with the maximum contact pressure observed in opposite corners. From the instant (E) the analytical and numerical solutions tend to overlap again. Although globally in agreement, the numerical solution represents the impact in a smoother way than the analytical one, the latter based on the hypothesis of only two potential contact points. The effect of the time step on the present solution is shown in Fig. 4where the previously selected time step Dt¼0:013 s and the analytical solution are compared with different time steps, i.e. Dt¼0:026 s and Dt¼0:007 s. As it can be noted, the time step has a direct effect on the energy dissipation, leading to normalized rocking angle time histories sensibly different (Fig. 4a). This effect is clearly shown in Fig. 4b, where the normalized total energy time histories show larger energy drops at impacts for larger time steps. Interestingly, the half rocking period versus normalized rocking angle diagram for subsequent impacts shown in Fig. 4c highlights that the present modelling strategy is able to accurately represent the nonlinear period-to-amplitude relationship as in Eq. (3), independently from the time step utilized. Indeed, the time step only rules the energy losses at impacts and therefore the distance between the points in the period-to-amplitude diagram. Consequently, the three cases considered in Fig. 4c lay on the same curve. As in the classical rocking theory the distance between the points in the period-to-amplitude diagram is governed by the coefficient of restitution in Eq. (2), it appears that the time step could be tuned to guarantee the desired energy dissipation at impacts. 123 Rocking block simulation based on numerical dissipation The free rocking response of the same block for different values of initial amplitude (h0=a) with the same Dtis compared with the analytical solution given by Eq. (3) in Fig. 5, in terms of normalized rocking angles (Fig. 5a) and half rocking periods (Fig. 5b), along with the number of impacts. As it can be observed, the problem appears amplitude independent, i.e. the same Dtcan be utilized independently of the rocking angle. This feature appears particularly appealing, and guarantees a reasonable generalization of the present computational approach. As a result, the surgical use of the numerical dissipation of the time integration scheme allows the modelling of energy dissipation at impacts in a phenomenological manner, allowing fast numerical simulations. In other words, the proposed modelling strategy permits to account for the desired energy dissipation while using the largest possible time step. 3 Setting of the time step In this section, a strategy for the setting of the time step Dtto guarantee good estimates of the rocking response based on an extensive numerical campaign and a multivariable nonlinear regression is discussed. The coefficient of restitution ris here assumed as an independent parameter [57]. The choice to keep r independent allows the employment of any experimentally measured coefficient of restitution (e.g. [33,65,71]) or theoretically improved model (e.g. [47,72–74]) to be adopted, which might differ from the one in Eq. (2)[33,64,65]. Furthermore, it has been found from preliminary analyses that the setting of the time step Dtis influenced, beyond the contact stiffness kn, as discussed in Section 2.2, and the coefficient of restitution r[57], by the size and the aspect ratio of the block, i.e. by Rand H=B, respectively. Accordingly, the setting of the time step might be reasonably represented, Fig. 3 Phase portrait for the case shown in Fig. 2. Comparison between the analytical and the present solutions (left). Magnified phase portrait at the first impact (top right). Contact pressure contour plots, from red (maximum contact pressure) to blue (no contact), at subsequent instants 123 A. M. D’Altri et al. similarly to what utilized in [57] for an akin problem, by the following simple function: Dt¼A1RH B  A2 kA3 nln rðÞ ð6Þ where A1,A2, and A3are coefficient to be determined. It is here highlighted that such function will provide Dt¼0 for r¼1. This extreme case is out of interest for the present study as r\1 for all real cases. A suitable strategy for the identification of A1,A2, and A3 is discussed in the following. Fig. 4 Effect of the time step on the present solution based on numerical dissipation (see Fig. 2for the settings). Comparison of the present solution (Dt¼0:013 s) with two other time steps (Dt¼0:026 s and Dt¼0:007 s) in terms of anormalized rocking angle time history, bnormalized total energy time history, and chalf rocking period versus normalized rocking angle diagram for subsequent impacts Fig. 5 Free rocking response for different values of initial amplitude (h0=a) with the same Dt, in terms of anormalized rocking angles and bhalf rocking periods, along with the number of impacts. Solid lines represent the analytical solution, while hollow circles represent numerical solutions 123 Rocking block simulation based on numerical dissipation 3.1 Numerical campaign An extensive numerical campaign is here carried out to estimate the coefficients A1,A2, and A3in Eq. (6). In total, 10 different block geometries with 3 different aspect ratios are considered, as specified in Table 1. Such geometries are chosen to have a reasonable replication of real rocking structures (namely cultural heritage structures). For each block geometry, 4 different values of knare considered, i.e. 2.5e ?08, 5e ?08, 10e ?08, and 25e ?08 N/m 3 (for later convenience labeled as k2.5, k5, k10, and k25, respectively), adopted in a reasonable range according to [57,75]. Finally, for each case, 40 values of Dtare considered within the range of 0.001 s, 0.002 s, …., 0.040 s. Accordingly, the numerical campaign here discussed is composed of 1600 numerical simulations of free rocking. Given the amplitude independence shown in Fig. 5, all these simulations have been conducted by adopting h0=a¼0:5. According to [64,76], a material density equal to 2600 kg/m 3 has been here adopted for the blocks, while the density variability for common stone/masonry construction materials is consider negligible for the calibration purposes of this work. In all cases, the free rocking response is analysed for 30 s, or the minimum time needed to reach the maximum rocking angle of a cycle equal to 0:05h0=aif greater than 30 s. To visualize the various blocks geometries considered, their proportions are highlighted in Fig. 6, organized with the same layout of Table 1. 3.2 Regression for the setting of the time step The comparison against the analytical solution in Eq. (3) is performed in terms of rocking angle and half rocking period, for a significant number of impacts N, which is here identified as the number of impacts needed to reach a rocking angle lower than 0:05h0=a. An example of comparison between analytical and numerical solutions is shown in Fig. 7, in terms of normalized rocking angle (Fig. 7a) and half rocking period (Fig. 7b), along with the number of impacts for a certain rand a certain Dt. An error measurement between analytical and numerical solutions is here introduced. Firstly, the root-mean-square error of the rocking angle normalized on the initial angle (eh=a), and of the half rocking periods normalized on the initial period (eT=2)is computed, for a significant number of impacts N, as: eh¼1 b h0 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 NX N n¼0b hne hn  2 v u u t; eT¼1 b T0=2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 NX N n¼0b Tn=2e Tn=2  2 v u u t ð7Þ where the symbol ^ denotes values obtained through the analytical solution, and the symbol ~ denotes values obtained through the numerical solution. Table 1 Blocks geometries employed in the numerical campaign Block label 2H[m] 2B[m] R[m] H=B[–] HB4_R0.62 1.2 0.3 0.62 4 HB4_R1.24 2.4 0.6 1.24 4 HB4_R2.48 4.8 1.2 2.48 4 HB4_R3.71 7.2 2.8 3.71 4 HB7_R1.06 2.1 0.3 1.06 7 HB7_R2.12 4.2 0.6 2.12 7 HB7_R4.24 8.4 1.2 4.24 7 HB10_R1.51 3.0 0.3 1.51 10 HB10_R3.02 6.0 0.6 3.02 10 HB10_R6.04 12.0 1.2 6.04 10 Fig. 6 Blocks geometry proportions used in the numerical campaign. For actual sizes, refer to Table 1 123 A. M. D’Altri et al. Secondly, these two error measures are combined into a unique global measurement of the relative error eG, defined as: eG¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 hþe2 T qð8Þ This global measurement of the relative error eGis here used within a sort of optimization problem, where the optimal Dtis selected as the case with the lowest eG, aimed at investigating the optimal Dtto be used in numerical simulations given the block geometry, r, and kn. Accordingly, eGis computed for each of the 40 considered Dts, for each block listed in Table 1, for each value of contact stiffness kn, and for coefficients of restitution rvarying within the range 0.80, 0.81, …, 0.99. Examples of distribution of eGalong with the considered time steps are shown in Fig. 8, where the error trends between two different values of coefficient of restitution (a lower value 0.88, and a higher value 0.95) are compared. As it can be noted, the error eGreaches small values for a range of Dts. In particular, it is observed that lower values of r(e.g., 0.88) show smoother trends of eG, i.e., a wider range of Dtis characterized by small errors, while higher values of r(e.g., 0.95) show steeper trends of eG, i.e., a narrower range of Dtis characterized by small errors. By way of example, with reference to Fig. 8, an error eG10%is obtained with 0:10 sDt0:17 sfor r¼0:95, while with 0:26 sDt0:40 sfor r¼0:95 (considering also that values of Dt[0:40 shave not been here considered). Accordingly, the adoption of a Dtin a neighborhood of the optimal Dtwould still guarantee accurate results. This is particularly true for lower values of coefficient of restitution, i.e. for values of rexpected in real historical structures with, e.g., mortar joints and/or defects. Moreover, it should be pointed out that the present approach is based on the HHT method and the convergence within each increment is not guaranteed. Indeed, for highly nonlinear problems (e.g. damage constitutive laws, contact cohesion, etc.) convergence may not be found within a time increment. This might be overcome by reducing the time increment (only for the non-converged increment, e.g., by 50%) to obtain a solution. This aspect may affect locally (in time) the dissipative properties of the solution, and the analysis report should be checked to judge the quality of the response. In any case, in all the simulations considered in this paper, non-converged increments have not been recorded. Fig. 7 Example of comparison between analytical and numerical solutions, in terms of anormalized rocking angle and bhalf rocking period, along with the number of impacts, for r = 0.89 and Dt=0.009 s (case HB4_R0.62_k2.5) Fig. 8 Distribution of eGalong with the considered time steps: examples of a lower value (r¼0:88) and a higher value (r¼0:95) of coefficient of restitution (case HB7_R2.12_k2.5) 123 Rocking block simulation based on numerical dissipation computational approach, a rocking block has been idealized as a solid body interacting with its foundation through a contact-based formulation. The wellknown HHT method, set to optimally treat dissipation in contact problems, has been employed, being the numerical dissipation governed by the time step. The rocking dissipative phenomenon at impacts appeared to be accurately predicted by the proposed computational approach without the use of any damping model. A broad numerical campaign has been conducted to define a regression law in analytic form for the setting of the optimal time step. Such law has been found to be dependent on the block size and aspect ratio, the contact stiffness, as well as the coefficient of restitution selected. The so-obtained regression law appeared accurate and an a posteriori validation with cases not in the training dataset confirmed the effectiveness and the robustness of the approach. Interestingly, it has been found that by increasing the block size also the optimal time step increases (so allowing fast dynamic Fig. 17 Comparison with the experimental tests by Bachmann et al. [15]. aFree rocking normalized rocking angle time history, adapted from [15]. bNormalized rocking angle and chalf rocking period along with the number of impacts. dEarthquakelike input, Case Lefkada 2H = 10 m, cumulative distribution functions of the maximum normalized rocking angle for 100 tests (adapted from [15]) 123 A. M. D’Altri et al. simulations even for large-scale structures). In particular, it has been found that rocking blocks with sizes of interest for structural engineering (namely cultural heritage structures) can be simulated with time steps within 10 –3 710 –1 s, so allowing very fast computations. Finally, the comparison with available experimental tests highlighted the efficacy of the present computational approach for free rocking and harmonic loading cases (in a deterministic sense), and for earthquake-like loading cases (in a statistical sense, i.e., in terms of cumulative distribution functions). Future developments will concern the extension of the present computational approach to multi-block rocking structures, e.g., by exploiting the concept of dynamically equivalent rocking structures [23] to set the time step in a straightforward way. Acknowledgements The authors wish to thank Prof. Fernando Pen ˜a from Universidad Nacional Auto ´noma de Me ´xico, Mexico City, Mexico, for providing the experimental data of Pen ˜a et al. (2008), and Prof. Michalis Vassiliou from ETH, Zurich, Switzerland, for providing the experimental data of Bachmann et al. (2008). Their support is gratefully acknowledged. Author contributions AMD: Conceptualization, Methodology, Investigation, Software, Writing – original draft, Funding acquisition; GV: Methodology, Software, Writing – review & editing; SdM: Supervision, Resources, Writing – review & editing; PBL: Supervision, Writing – review & editing, Funding acquisition. Funding Open access funding provided by Alma Mater Studiorum - Universita `di Bologna within the CRUI-CARE Agreement. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 101029792 (HOLAHERIS project, ‘‘A holistic structural analysis method for cultural heritage structures conservation’’ https://site.unibo.it/holaheris/en). This study has been also funded by the STAND4HERITAGE project (new STANDards FOR seismic assessment of built cultural HERITAGE) that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant No 833123) as an Advanced Grant. The second author is financed by national funds through Portuguese FCT – Foundation for Science and Technology, under grant agreements 2020.07325.BD. Data availability The datasets analyzed during the current study are available in an open data repository. 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